The strength of the second harmonic electric field, E2ω, that is produced at charged interfaces is a function of the incident fundamental electric field, Eω, the second order susceptibility of the interface, χ(2), the zero-frequency electric field corresponding to the interfacial potential produced by surface charges, Φ(0), and the third-order susceptibility, χ(3), according to1-6 the following equation:√{square root over (ISHG)}∝E2ω∝χ(2)EωEω+χ(3)EωEωΦ(0)
Early work, in which the relative phase of the terms contributing to the second harmonic generation (SHG) intensity was included7, shows that when the wavelength of the fundamental and second harmonic photons are far from electronic and vibrational resonance, χ(2) and χ(3) are real, though they may differ in sign7,8. Yet, phase information has not been recovered in traditional SHG detection schemes, as they only collect the square modulus of the signal. While phase information from SHG and vibrational sum frequency generation (SFG) signals can be obtained through coherent interference of the signal of interest with an external9-16 or internal7,18 phase standard, applications of such reference techniques to determine the phase of SHG signals generated at buried interfaces, such as charged oxide/water interfaces, is challenging due to the presence of dispersive media on both sides of the interface. Additionally, the interface between water and α-quartz, the most abundant silicate mineral in naturel19-21, has been theoretically predicted to produce a more ordered interfacial water layer than amorphous silica22-24, though this has not yet been probed using even traditionally detected SHG, as the noncentrosymmetric bulk generally produces second harmonic signals that overpower surface SHG signals by orders of magnitude to the point where the surface signal is indistinguishable from the bulk response.